First-Order Logic. Doug Downey Northwestern EECS 348 Intro to AI Based on slides by Stuart Russell

Transcription

1 First-Order Logic Doug Downey Northwestern EECS 348 Intro to AI Based on slides by Stuart Russell

2 Pros and Cons of Propositional Logic Declarative: pieces of syntax correspond to facts Allows partial/disjunctive/negated info Unlike most data structures and DBs Compositional: Meaning of A B derived from meanings of A, B Meaning is context-independent Very limited expressiveness Can t say pits cause breezes in adjacent squares except by writing a sentence for each square

3 First-order Logic Objects Greg, Sheridan Road, 25, blue, Moby Dick (the novel), Relations Tall, four-lane, perfect square, has color, has read, Functions Height, father of, square root,

4 FOL: Syntax

5 Atomic Sentences

6 Complex Sentences Formed from simple sentences using connectives: E.g., MayorOf(Daley, Chicago) => LivesIn(Daley, Chicago) Eat(Doug, Nachos) Eat(Doug, Grapes)

7 Truth in FOL A model in FOL is a set of objects, and full definitions for relations and functions Model enumeration totally infeasible For each relation (of arity k) For each distinct subset of all combinations of k objects Technically, objects/relations in model are mapped to particular KB symbols through an interpretation This provides yet more complexity

8 Universal Quantification <variables> <sentence> Everyone at Northwestern is smart: x At(x, Northwestern) => Smart(x) x P is true in a model m if P is true with any object from m substituted for x in P

9 Common Mistake to Avoid Typically => is the main connective with Don t use by mistake, e.g.: x At(x, Northwestern) Smart(x)

10 Existential Quantification <variables> <sentence> Someone at Northwestern is rich: x At(x, Northwestern) Rich(x) x P is true in a model m if P is true with some object from m substituted for x in P

11 Common Mistake to Avoid Typically is the main connective with Don t use => by mistake, e.g.: x At(x, Northwestern) => Rich(x) True if anyone is not at Northwestern!

12 Properties of Quantifiers x y is the same as y x x y is the same as y x x y is not the same as y x Related through negation: x Likes(x, IceCream) x Likes(x, IceCream) x Likes(x, Broccoli) x Likes(x, Broccoli)

13 Examples! Brothers are siblings x y Brother(x, y) => Sibling(x, y) Sibling is symmetric: x y Sibling(x, y) => Sibling(y, x) One s mother is one s female parent x y Mother(x, y) <=> (Female(x) Parent(x, y)) A first cousin is a child of a parent s sibling

14 Payoff in the Wumpus World FOL is concise Can define Adjacent(x, y) for squares x, y Then: y Breezy(y) [ x Pit(x) Adjacent(x, y)]

15 Interlude: Recap and RoadMap So far classical AI Overview and Philosophy of AI (e.g., Turing Test) General search methods and techniques A*, Local Search, CSPs, etc. Logic Logical Agents, Propositional Logic, FOL (inference: today) To Come modern AI Challenges for classical AI Leveraging data (machine learning) The big questions

16 Interacting with FOL KBs Say a wumpus-world agent perceives a smell and a breeze, but no glitter, at t = 5: I.e., does the KB entail any actions at t = 5? Answer: Yes, {a / Shoot} <= substitution Ask(KB, S) returns the substitutions of S that KB entails

17 Inference in first-order knowledge We just said: bases Ask(KB, S) returns the substitutions of S that KB entails How? Propositionalize Instantiation Forward/Backward Chaining Resolution Key Idea: Unification

18 Resolution use in practice Theorem provers Provided major mathematical results Otter proved a conjecture (the Robbins algebra) which had been unsolved for 60 years Verification Hardware (adders, CPUs) Algorithms RSA encryption Software Spacecraft control

19 Logic as a Foundation for AI Logic: extremely expressive, powerful Theorem provers: useful in practice But: Writing down needed knowledge is hard So-called Frame, qualification, ramification problems => Knowledge acquisition bottleneck Logic systems are incomplete Logic systems are brittle

20 The real world: Sensing and Acting Perception three binary inputs [smell, breeze, glitter] at each time t s, b, t Percept([s, b, Glitter], t] => AtGold(t) t AtGold(t) => Action(Grab, t)? Infinite Loop! t AtGold(t) Holding(Gold, t) => Action(Grab, t)

21 Keeping track of Change Facts hold in particular situations E.g., Holding(Gold, t) may be False, Holding(Gold, t+8) true Agent must keep track of change

22 Frame Problem Effect axioms t Standing( (i, j), t) Facing(Up, t) Action(Forward, t) => Standing( (i,j+1), t + 1) But HaveArrow(t + 1)? Frame axioms keep track of what doesn t change Action(Forward, t) => (HaveArrow(t) HaveArrow(t + 1)) Etc. etc. etc.

23 Representational Frame Problem Historically thought to be extremely tricky Can be solved by writing axioms about fluents rather than actions Holding(Gold, t) <=> Holding(Gold, t-1) and action at t-1 made it true or Holding(Gold, t-1) and no action at t-1 made it false

24 Qualification Problem Action s preconditions can be complex Action(Grab, t) => Holding(t).unless gold is slippery or nailed down or too heavy or our hands are full or

25 Ramification Problem Actions can have many consequences t Standing( (i, j), t) Facing(Up, t) Action(Forward, t) But also => Standing( (i,j+1), t + 1) => In( Basketball, (i, j+1), t + 1) if I m holding a basketball Writing all these down -- difficult

26 Knowledge Acquisition Remember the Colonel West story We converted text to logic In practice who does this? Qualification, Ramification problems tell us we need tons of common-sense knowledge The infamous knowledge acquisition bottleneck

27 Knowledge Acquisition: Options Type it all in yourself Cyc Get Web citizens to type it all in Open Mind Extract it from the Web KnowItAll, TextRunner, Fusion Tables, Wolfram Alpha, WikiData.

28 Logic as a Foundation for AI Logic: extremely expressive, powerful Theorem provers: useful in practice But: Writing down needed knowledge is hard So-called Frame, qualification, ramification problems => Knowledge acquisition bottleneck Logic systems are incomplete Logic systems are brittle

29 Gödel's Incompleteness Theorem Completeness Theorem: All valid statements have proofs in FOL Incompleteness Theorem: For any FOL KB enhanced to allow mathematical induction, there are true statements that can t be proved.

30 Gödel's Theorem: Sketch (1) Idea: This statement is false. More specifically: This statement has no proof.

31 Gödel's Theorem: Sketch (2) Assign numbers to sentences, proofs E.g. by sorting by length, then alphabetically Consider the sentence (j, A) For all numbers i, statement #i is not a proof for statement #j from the axioms A Let be the sentence (#, A) false? But it has a proof! true? It s unprovable!

32 Gödel's Theorem: Ramifications Argument: Computers are limited by Gödel's theorem, whereas humans aren t. Thus, AI is doomed

33 Three counter-arguments Gödel's theorem applies to math induction systems, e.g. Turing Machines Computers aren t really Turing machines Steve cannot say this sentence is true. But Steve might be able to do other cool stuff Are humans really immune to the theorem?

34 Logic as a Foundation for AI Logic: extremely expressive, powerful Theorem provers: useful in practice But: Writing down needed knowledge is hard So-called Frame, qualification, ramification problems => Knowledge acquisition bottleneck Logic systems are incomplete Logic systems are brittle

35 Brittleness of Logic Systems Consider a KB with just one contradiction That KB entails everything This is a problem because much of the world is uncertain Perception, action, incomplete information, controversies, etc.

36 Toward Modern AI Limitations: Knowledge Acquisition Bottleneck, Brittleness Modern directions: Situatedness, embodiment Probability Learning from data Combining the above with logic

37 Alternatives: Focus on Behavior Argument: we can t even build systems that do what ants do In the timeline of evolution, simple cells->ants took much longer than ants->humans Let s start by building ants Environment, body can make tasks easier Incrementally solve real problems end-to-end

38 Intelligent Agents Sensory/motor aspect more important, more coupled, more integrated with rest of intelligence than originally thought

39 Behavior-based robots as a foundation for AI Common-sense knowledge arises from our interaction in the world Thus, the road to AI is paved with real-world interaction We must build robots Another possibility: softbots

40 Subsumption Architecture Behavior-based robotics Beam-wiki.org

41 Other modern trends Biological inspiration, e.g.: Neural networks Hexapod robots drawing on insect nervous systems followed subsumption architecture Probability theory Handles uncertainty, overcomes brittleness Data

42 Learning from Data Quantities of data are exploding -- let s learn from it Machine learning